Numerical solution of Differential Equations & Application Area
Numerical solution of Differential Equations & Application Area
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Lab4B (Version 2)
Numerical solution of Differential Equations & Application Area
In Lab4B will explore methods for solving ODE numerically and apply these methods. Sources for this Lab
include sections 6.1, 6.2 and parts of 6.3 and 6.4. You are also encouraged to find your own sources.
Some additional resources will be made available. Labs will go ahead as usual this week and the
following week.
You will write an integrated essay Report, in the same format as the previous labs. The report +
appendices etc should be submitted to Gradescope. It should cover the following aspects.
Title, team members, Abstract, Introduction (max 1 page)
Section 1 Introduction
Section 2 Numerical Solution of ODE (max 2 pgs)
This section is about the numerical solution of ODE. It should include a brief discussion of the main
methods you encountered, accuracy and why the methods are important, and be illustrated by
examples. As usual you should integrate a storyline that connects this section with the rest of the
report, and your chosen focus in Section 3.
Section 3 Application of Numerical Solution of ODE to [You choose one of Q3, Q4, Q5, Q6, Q7]
For completeness the Problem Areas Q3 to Q5 are outlined here. As updates are made to these
Problem Areas, they will be noted in revised versions of this document.
3-Q3: Improvements of Newton’s Method with application to Fractals
Part of this question is developing more fractal graphics, extending your earlier lab work. This involves producing
fractal graphics more complicated polynomials and improving Newton’s method to make better graphics.
One aspect that will be improved for Newton’s method is guaranteeing it numerically approximates all n roots of
an n-th degree polynomial p(z) = 0. This method starts from the n known roots of an easily exactly solvable related
polynomial, then numerical solution of an ODE is executed to numerically follow paths from the known roots to all
n roots of the difficult polynomial. In particular, start with the easy polynomial q(z) = z^n – 1 = 0 and give the roots
in terms of complex exponentials (see an earlier lab) for n = 2 and n=3. Suppose that p(z) = 0 is the difficult
polynomial and you want to approximate all the roots of p(z) = 0. Then let H(t) = (1-t)*(z(t)^n – 1) + t*p(z(t)) = 0.
Show that H(0) = z^n – 1, and H(1) = p(z). Show that the derivative of H(t) gives
H’(t) = ( (1-t)*n*z(t)^(n-1) + t*p’(z(t)))*z’(t) + p(z(t))- (z(t)^n – 1) = 0 (H-Eqn)
Then solve for z’(t) to show that you get
z’(t) = ((z(t)^n – 1) - p(z(t)))/((1-t)*n*z(t)^(n-1) + t*p’(z(t))) = F(t,z(t)) (H-DEqn)
Then to approximate all the roots of p(z) = 0 use a numerical ODE solver for t = 0 .. 1 with initial conditions z(0)
equal to each of the exactly known roots of q(z) = z^n – 1 = 0. The output at t=1 are the desired roots. Give
examples for n = 2, 3 and higher if you can.
Though quite simple this intersects with research of Siyuan Deng, Greg Reid the Math Dept’s new hire
Taylor Brysiewich. [I suspect that he changed his name to Taylor since he is researching Taylor-Newton methods!].